Nikolai Filonov, Frederic Klopp
We prove that the spectrum of a Schrodinger operator that is periodic in certain directions and super-exponentially decaying in the others is purely absolutely continuous.
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We prove estimates on the Hölder exponent of the density of states measure for discrete Schrödinger operators with potential of the form $V(n) = λ(\lfloor(n+1)β\rfloor - \lfloor nβ\rfloor)$, with $λ$ large enough, and conclude that for almost all values of $β$, the density of states measure is not Hölder continuous.
We consider an alloy type potential on an infinite metric graph. We assume a covering condition on the single site potentials. For random Schrödingers operator associated with the alloy type potential restricted to finite volume subgraphs we prove a Wegner estimate which reproduces the modulus of continuity of the single site distribution measure. The Wegner constant is independent of the energy.
We consider the Schrödinger operator in ${\mathbb R}^n$, $n\geq 3$, with the electric potential $V$ and the magnetic potential $A$ being periodic functions (with a common period lattice) and prove absolute continuity of the spectrum of the operator in question under some conditions which, in particular, are satisfied if $V\in L^{n/2}_{\mathrm {loc}}({\mathbb R}^n)$ and $A\in H^q_{\mathrm {loc}}({\mathbb R}^n;{\mathbb R}^n)$, $q>(n-1)/2$.
Quantum waveguide with the shape of planar infinite straight strip and combined Dirichlet and Neumann boundary conditions on the opposite half-lines of the boundary is considered. The absence of the point as well as of the singular continuous spectrum is proved.
For analytic quasi-periodic Schrödinger cocycles, You and Zhang [9] proved that the Lyapunov exponent is Hölder continuous for weak Liouville frequency. In this paper, we prove that the Hölder continuity also holds if the analytic potential is weakened to Gevrey potential.
Stability of the non-critical spectral properties I: arithmetic absolute continuity of the integrated density of states
本文证明了在特定算术频率条件下，非临界几乎 Mathieu 算子的频率无关解析扰动保持态密度绝对连续性。
In this paper, for d > 2, we prove the absolute continuity of the spectrum of a d-dimensional periodic Dirac operator with some discontinuous magnetic and electric potentials. In particular, for d = 3, electric potentials from Zygmund classes $L^3\ln ^{1+δ}L(K)$, $δ>0$, and also ones with Coulomb singularities, with constraints on charges depending on the magnetic potential, are admitted (here K is the fundamental domain of the period lattice).
Anderson localization and Hölder continuity of the integrated density of states for analytic quasiperiodic Schrödinger operators
本文研究了准周期薛定谔算子在扰动范围内，对任意非常数解析势和任意迪ophantine频率的安德森局域化及积分态密度的霍尔德连续性，提出了一种新的格林函数控制方法以消除双重共振。
The paper contains a brief description of a simplified version of A. Sobolev's proof of absolute continuity of spectra of periodic magnetic Schrödinger operators. This approach is applicable to all periodic elliptic operators known to be of interest for math physics (including Maxwell), and in all these cases leads to the same model problem of complex analysis. The full account of this approach will be provided elsewhere.
We prove that the Floquet spectrum of a class of time-periodic Schroedinger equations under a a mildly nonlinear resonant forcing is purely absolutely continuous.
We prove that 3-dimensional Schrodinger operator with slowly decaying potential has an absolutely continuous spectrum that fills the positive half-line. The asymptotics of Green's function is obtained as well.
We study discrete quasiperiodic Schrödinger operators on $\ell^2(\zee)$ with potentials defined by $γ$-Hölder functions. We prove a general statement that for $γ>1/2$ and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of the upper Lyapunov exponent of strictly ergodic cocycles.
We prove that the spectrum of the discrete Schrödinger operator on $\ell^2(Z^2)$, $(ψ_{n,m})\mapsto -(ψ_{n+1,m} +ψ_{n-1,m} +ψ_{n,m+1} +ψ_{n,m-1})+V_nψ_{n,m}$ is absolutely continuous.
We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schrödinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).
We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrodinger operators on the strip and on the Caley tree (Bethe lattice) are considered.
We consider a model of leaky quantum wire in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction.
We consider a family of random Schrödinger operators on the discrete strip with decaying random $\ell^2$ matrix potential. We prove that the spectrum is almost surely pure absolutely continuous, apart from random, possibly embedded eigenvalues, which may accumulate at band edges.
We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon's list of Schrödinger operator problems for the twenty-first century.
We show that a generic quasi-periodic Schrödinger operator in $L^2(\mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operator with the resulting potential has empty absolutely continuous spectrum.
We prove new criteria of stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and periodic Schrödinger operators is preserved under perturbations by all potentials $V(x)$ satisfying $|V(x)| \leq C(1+x)^{-\frac{2}{3}-ε}.$ The main new technique includes an a.e.\ convergence theorem for a class of integral operators.
Using a perturbative argument, we show that in any finite region containing the lowest transverse eigenmode, the spectrum of a periodically curved smooth Dirichlet tube in two or three dimensions is absolutely continuous provided the tube is sufficiently thin. In a similar way we demonstrate absolute continuity at the bottom of the spectrum for generalized Schrödinger operators with a sufficiently strongly attractive $δ$ interaction supported by a periodic curve in $\mathbb{R}^d, d=2,3$.
Consider $Γ$, a non-degenerate lattice in $\R^2$ and a constant magnetic field $B$ with a flux though a cell of $Γ$ that is a rational multiple of $2π$. We prove that for a generic $Γ$-periodic potential $V$, the spectrum of the Landau Hamiltonian with magnetic field $B$ and periodic potential $V$ is purely absolutely continuous.
We consider a family of operators $-Δ+ t V$ with a slowly decaying and oscillating potential $V$. We prove that the absolutely continuous spectrum of this operator is essentially supported by $[0,\infty)$ for almost every $t$.
We consider a family of multi-dimensional Schrödinger operators $-Δ+t V$ with a real $t$. The potential $V$ in our model decays at infinity in a special way, so that it satisfies a certain integral condition. We prove that the absolutely continuous spectrum of this operator covers the interval $[0,\infty)$ for almost every $t$.
Hölder continuity and Fourier asymptotics of spectral measures for 1D Schrödinger operators under decaying perturbations
本文研究了一维Schrödinger算子在指数衰减扰动下的谱测度连续性与量子返回概率渐进行为，证明了谱测度的1/2-Hölder连续性和纯绝对连续谱的持续性。
Consider the Dirichlet Laplacian operator $-Δ^D$ in a periodic waveguide $Ω$. On the condition that $Ω$ is sufficiently thin, we show that its spectrum $σ(-Δ^D)$ is absolutely continuous (in each finite region). In addition, we ensure the existence of at least one gap in $σ(-Δ^D)$ and locate it.
The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic Schrödinger operators is preserved under all perturbations $V(x)$ satisfying $|V(x)|\leq C(1+x)^{-α}$, $α>\frac{1}{2}.$ This result is optimal in the power scale. More general classes of perturbing potentials which are not necessarily power decaying are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schr…
The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr{ö}dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density…
Neat stuff about eigenfunctions, transfer matrices, and a.c. spectrum of one-dimensional Schrodinger operators
The objects of the present study are one-parameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multi-dimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-contin…
We show that the law of the KPZ fixed point starting from arbitrary initial condition is absolutely continuous with respect to the law of Brownian motion $B$ on every compact interval. In particular, the Airy$_1$ process is absolutely continuous with respect to $B$ on any compact interval.
The concept of the generalized continuity equation (GCE) was recently introduced in [J. Phys. A: Math. and Theor. {\bf 52}, 1552034 (2019)], and was derived in the context of $N$ independent Schrödinger systems. The GCE is induced by a symmetry transformation which mixes the states of these systems, even though the $N$-system Lagrangian does not. As the $N$-system Schrödinger Lagrangian is not invariant under such a transformation, the GCE will involve source terms which, under certain conditions vanish and lead to conserved currents. These conditions may hold globally or locally in a finite d…
We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on $L^2 (\R^d)$, for $d \geq1$, is locally Hölder continuous at all energies with the same Hölder exponent $0<α\leq1$ as the conditional probability measure for the single-site random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The single-site potential $u\in L\_0^\infty (\R^d)$ must be nonnegative and compactly-supp…
We prove a dichotomy between absolute continuity and singularity of the Ginibre point process $\mathsf{G}$ and its reduced Palm measures $\{\mathsf{G}_{\mathbf{x}}, \mathbf{x} \in \mathbb{C}^{\ell}, \ell = 0,1,2\dots\}$, namely, reduced Palm measures $\G_{\mathbf{x}}$ and $\G_{\mathbf{y}}$ for $\mathbf{x} \in \mathbb{C}^{\ell}$ and $\mathbf{y} \in \mathbb{C}^{n}$ are mutually absolutely continuous if and only if $\ell = n$; they are singular each other if and only if $\ell \not= n$. Furthermore, we give an explicit expression of the Radon-Nikodym density $d\G_{\mathbf{x}}/d \G_{\mathbf{y}}$ fo…
$L^p$-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions
本文研究高维高阶薛定谔算子在阈值特征值存在时波算子的 $L^p$ 有界性，统一处理奇偶维数，并将有界范围推广至与经典二阶情形一致。
Recently, Hislop and Marx studied the dependence of the integrated density of states on the underlying probability distribution for a class of discrete random Schrödinger operators, and established a quantitative form of continuity in weak* topology. We develop an alternative approach to the problem, based on Ky Fan inequalities, and establish a sharp version of the estimate of Hislop and Marx. We also consider a corresponding problem for continual random Schrödinger operators on $\R^d$.
It is proven that the absolutely continuous spectrum of matrix Schrödinger operators coincides (with the multiplicity taken into account) with the spectrum of the unperturbed operator if the (matrix) potential is square integrable. The same result is also proven for some classes of slower decaying potentials if they are smooth.
On the dependence of the nonlinear Schrodinger flow upon the power of the nonlinearity
本文研究散焦能量次临界幂型 NLS 流随非线性幂次的连续性，建立局部/全局时间连续、重标度模平方的一致收敛及幂次趋于零时向对数 NLS 的收敛。
Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schrödinger Operators
本文研究了多维准周期Schrödinger算子的绝对连续谱，证明了一般算子的谱包含半轴，构造了本征函数，并基于动量空间多尺度分析完成证明。
We prove results about the Hölder continuity of the spectral measures of the extended CMV matrix, given power law bounds of the solution of the eigenvalue equation. We thus arrive at a unitary analogue of the results of Damanik, Killip and Lenz about the spectral measure of the discrete Schrödinger operator.
In this paper we prove stable determination of an inverse boundary value problem associated to a magnetic Schrödinger operator assuming that the magnetic and electric potentials are essentially bounded and the magnetic potentials admit a Hölder-type modulus of continuity in the sense of $L^2$.
We prove that the the density of states measure (DOSm) for random Schrödinger operators on $\mathbb{Z}^d$ is weak-$^*$ Hölder-continuous in the probability measure. The framework we develop is general enough to extend to a wide range of discrete, random operators, including the Anderson model on the Bethe lattice, as well as random Schrödinger operators on the strip. An immediate application of our main result provides quantitive continuity estimates for the disorder dependence of the DOSm and the integrated density of states (IDS) in the weak disorder regime. These results hold for a general …
We review a geometric approach to proving absolutely continuous (ac) spectrum for random and deterministic Schrödinger operators developed in \cite{FHS1,FHS2,FHS3,FHS4}. We study decaying potentials in one dimension and present a simplified proof of ac spectrum of the Anderson model on trees. The latter implies ac spectrum for a percolation model on trees. Finally, we introduce certain loop tree models which lead to some interesting open problems.
Exact local distribution of the absolutely continuous spectral measure
该研究拓展了单频薛定谔算子谱测度连续性的相关结论，精确刻画了稠密小势场下谱测度的局部分布，并探究了其在次临界能量下的分层赫尔德连续性。
Classes of the nonlinear Schrodinger-type equations compatible with the Galilei relativity principle are described. Solutions of these equations satisfy the continuity equation.
The absolutely continuous spectrum of an ergodic family of one-dimensional Schrödinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absolutely continuous spectrum, the presence of absolutely continuous spectrum, and even the presence of purely absolutely continuous spectrum. We review these results and their recent applications to a number of problems: the absence of absolutely continuous spectrum for rough potentials, the absence of absolute…
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